CAPÍTULO II MANDO Y CONTROL
H. ACCIONES TÁCTICAS FUNDAMENTALES
3) Ataque Frontal
Throughout this chapter we will deal with the consequences of AS in the extended TO induced in fermionic Abelian theories by virtue of superconductivity using the language of an effective Chern Simons gauge theory. While this is attractive because it is amenable to generalization, in this section we concentrate on non-interacting systems for which we can write down explicit Hamiltonians. We will show numerically by
bandstructure calculations a level crossing in the Caroli-de Gennes-Matricon energy states in the vortex and corresponding fermion parity flip in some simple examples.
We first trace the conceptual origin of vortex fermion parity flip to the fermion parity pump in one- dimensional (1D) p-wave superconductors [Kitaev, 2001]. Having established the parity flip argument in 1D and the associated energy level crossing, we explore vortex parity flip in two dimensions (2D) in some instances which have been recently proposed as an exciting means of nucleating MBS.
As an explicit 1D example, the p-superconducting Kitaev wire represented by the lattice Hamiltonian [Ki- taev, 2001] H− µN =X r tc†rcr+1− µc†rcr+ ∆c†rc † r+1+ h.c. (4.5)
is topological and carries zero energy boundary MBS when the electron hopping strength|t| is bigger than the chemical potential |µ|. The low energy states of a superconducting ring with two weak links, one at r = 0 and the other at r = L/2, are labeled by the two local fermion parities (−1)F0 and (−1)FL/2.
Figure 4.2: A Kitaev p-wave ring with two links. The superconducting order parameter on the bottom ring is wound is wound from 0→ 2π. This flips the fermion parity at both the links.
When the phase of the pairing ∆ =|∆|eiφ winds adiabatically by 2π along a segment, say [0, L/2], there is a level crossing at each of the links accompanied by a fermion being pumped from one link to the other although there is superconducting pairing.
Towards generalizing the fermion parity pump argument to higher dimension, considering passing a quantum flux vortex across a proximity induced superconducting nanowire described by Eq.(4.5). The vortex brings spatial variation to the pairing phase ∆r=|∆|eiϕr, where ϕris the polar angle of site r from the vortex core. Figure 4.3 shows the level crossing of a 20-site system for t = |∆| = 2µ. When the flux vortex crosses the nanowire, a Bogoliubov - de Gennes (BdG) state on the nanowire is brought down to zero energy with a wavefunction localized at the point where the vortex intersects the wire. At the same time the fermion parity of the MBS pair flips. This mimics the fermion parity pump because the pairing phase
0 2̟ t-μ 0 E μ-t t+μ - t -μ s-superconductor
nanowire zero modes
flux vortex
Figure 4.3: Level crossing of a proximity induced superconducting nanowire when a hc/2e flux vortex passes across.
winds by 2π within the nanowire segment enclosed by the vortex trajectory. After a cycle the bulk nanowire is left with a fermionic excitation, which compensates for the parity flip of the MBS pair at the ends of the wire.
As we have seen, fermion parity flips have a general topological origin. Their presence is guaranteed by the change of sign of a single (or in general odd number of) MBS. In 2D, the BdG Hamiltonian H(k, θ, t) that describes the bulk superconducting medium around the MBS varies adiabatically by the polar angle θ centered at the MBS as a function of the adiabatic/temporal parameter t. The BdG Hamiltonian is topologically equivalent to a massive Dirac model
H(k, θ, t) = kxΓx+ kyΓy+ mΠ(θ, t) + O(k2) (4.6)
where Γx, Γy, Π(θ, t) are mutually anticommuting matrices and Γ2
x = Γ2y = 1. The model (4.6) having a slowly varying mass term unifies 2D fermion parity flip scenarios in different systems, two of which we now present. The first is a proximity induced superconducting (SC) interface between a quantum spin Hall insulator (QSH) [Kane and Mele, 2005a, Liu et al., 2008, K¨onig et al., 2007,Knez et al., 2014] and a trivial normal insulator (NI). The presence of protected zero energy MBS [Fu and Kane, 2008] requires time reversal (TR) breaking and can be facilitated by coating an (anti)ferromagnet (FM) along the interface (see figure4.4.
We take an 8-band square lattice model
H(r, t) = HQSH-NI(r)⊗ τz
+ ∆x(r, t)τx+ ∆y(r, t)τy+ h(r)µy (4.7)
HQSH-NI(r) = t(sin kxσx+ sin kyσy)µx
where σ, µ, and τ act on spin, orbital, and Nambu degrees of freedom, respectively. The Nambu basis is chosen to be (c↑,µ, c↓,µ, c†↓,µ,−c
†
↑,µ) so that (4.7) has a particle-hole symmetry Ξ = σyτyK, for K the complex conjugation operator. Eq.(4.8) describes the QSH-NI interface where the mass gap m(r) changes sign. We assume strong SC proximity so that the induced pairing order ∆ = ∆x+ i∆y is non-vanishing throughout the system. The antiferromagnet couples strongly to a strip neighborhood of the interface (see figure 4.4) where the FM order|h(r)| outweights the pairing |∆| but vanishes elsewhere.
The QSH-NI interface hosts a gapless 1D helical mode with opposite spins counter-propagating electrons. The helical mode is unstable to TR or charge conservation breaking perturbations. Its removal by magnetic field was seen in the earliest experiment of QSHI [K¨onig et al., 2007] and antiferromagnetic (FM) gapped edge was achieved in a graphene QSH state [Young et al., 2014]. Moreover, induced SC in QSH edge has been observed in HgCdTe quantum wells [Hart et al., 2014].
FM and SC are competing orders along the interface and a FM-SC domain wall – where |h| − |∆| changes sign – bounds a protected zero energy MBS. A pair of MBS are therefore located at the ends of the ferromagnet in figure4.4(a). The superconducting QSH-NI interface – except being TR symmetric and can only be realized holographically as the edge of a 2D system – can be treated as a Kitaev p-wave wire and thus carries protected boundary MBS. When a hc/2e flux vortex passes across the superconducting QSH-NI interface, it is akin to traveling across a Kitaev p-SC where there is a single protected level-crossing among the vortex states. This signals a vortex parity flip as the vortex is excited with one extra fermion after a cycle.
quantum spin Hall insulator (QSH)
trivial insulator (NI)
s-superconductor (SC) ferrom agne t (FM ) flux vortex 2! 0 E 0 |∆| |∆| 10 0 10 10 0 10 0.00 0.05 0.10 0.15 0.20 10 0 10 10 0 10 0.0 0.1 0.2 0.3
zero modes vortex state
(a)
(b) (c)
zero modes vortex state
t
Figure 4.4: Numerical result on a 32× 32 periodic lattice. A single level crossing when the vortex orbits the superducting quantum spin Hall-normal insulator-ferromagnet interface. Figures b) and c) show the magnitude of the probability associated with the wave functions of the zero mode and vortex respectively.
We have numerically verified the vortex parity flip phenomenon via its signature level crossing by putting the model (4.7) on a periodic 32×32 square lattice (see figure4.4(a)). The QSH-NI interface is located along the diagonal line and the four sides, which are sandwiched between the upper and lower triangular regions with opposite insulating mass m. We choose the hopping t = m, a uniform pairing strength|∆| = 0.5m and the antiferrormagnetic coupling h = 0.8m on a strip over half of the QSH-NI interface. To avoid monopole effects, we arrange a vortex and an anti-vortex with opposite flux, depicted by ⊗ and in figure 4.4(a). The vortices bring spatial and temporal variation to the SC pairing ∆(r, t) =|∆|eiϕ(r,t), for
eiϕ(r,t)= (z− w1(t))(z− w2)
|(z − w1(t))(z− w2)| (4.9)
where z = x + iy is the complex coordinates for lattice point r = (x, y), and wl = xl+ iyl are complex positions for the two vortex cores for l = 1, 2. The temporal dependence of (4.9) comes from the circular motion of the first vortex as it orbits around a MBS when t goes from 0 to 2π. The second vortex is kept stationary. The corresponding matrices in (4.6) in this system are
Γx= σxµxτz; Γy= σyµxτz;
Π(θ(t)) = mµzτz+ ∆xτx+ ∆yτy+ hµy (4.10)
Here, ∆ = ∆x+ i∆y =|∆|eiϕ is determined by (4.9) where w1(t) =|w1|eiθ(t). Figure4.4(a) shows a level- crossing of vortex states and confirms the fermion parity flip. At the crossing, a unit of fermion is pumped between the vortex and the MBS pair. Unlike the p-wave wire case (see figure 4.3) where the fermionic excitation is confined along the wire, here the excitation stays localized at the vortex as it moves away from the QSHI-NI interface until it is brought to the bulk bands and hybridizes with the rest of the system.
As another instance, vortex parity flip can also occur on proximity induced superconducting Chern insulators (CI) [Thouless et al., 1982, Haldane, 1988, Chang et al., 2013a] shown in figure 4.5. It can be described by the 4-band BdG Hamitlonian on a square lattice
H =t(sin kxσx+ sin kyσy)τz (4.11)
+ [m + (2− cos kx− cos ky)]σz+ ∆xτx+ ∆yτy
where σ, τ again act on spin and Nambu degrees of freedom similar to the previous case (4.7). Without the SC pairing, (4.11) describes an insulator with Chern number 1 when−2 < m < 0. The σz term is a TR breaking Zeeman coupling that competes with the induced s-wave pairing. We assume the pairing |∆| is
weaker than the insulating mass|m| so that the 2D system is not in the chiral p+ip phase [Qi et al., 2010] and a full quantum vortex does not hold a zero energy MBS. The consequence of the bulk Chern invariant is that
s-superconductor (SC) (a)
Chern insulator (CI) flux vortex s-superconductor - 0 2̟ |m-∆| 0 |m-∆| E 10 0 10 10 0 10 0.0 0.1 0.2 0.3 10 0 10 10 0 10 0.0 0.1 0.2 0.3
zero modes vortex state
(b) (c)
zero modes
vortex state
t
Figure 4.5: Numerical result on a 32× 32 periodic lattice. A single level crossing when the vortex orbits the end of the superconducting trench on the Chern insulator. Figures b) and c) show the magnitude of the probability associated with the wave functions of the Majorana zero mode pair and vortex state at the crossing.
the CI carries a gapless chiral edge mode that propagates in a single direction [Thouless et al., 1982,Haldane, 1988]. When two uncoupled CI’s with the same chirality are juxtaposed side by side, the interface bounds a pair of counter-propagating electron channels with opposite spins ψR↑, ψL↓. This gapless helical interface can be gapped out by TR breaking backscattering mψ†R↑ψL↓or U(1) breaking pairing ∆ψR↑† ψ
†
L↓between the two boundaries. These orders compete and a domain wall, where|m| − |∆| changes sign, traps a protected zero energy MBS. This can be realized by inserting a single-layer thick strongly superconducting trench in the 2D system, where |∆trench| > |m| (see figure4.5(a)). We again identify the matrices corresponding to (4.6) by
Γx= σxτz; Γy = σyτz; Π = mσz+ ∆xτx+ ∆yτy (4.12)
As with the superconducting QSH-NI interface considered above, the SC trench in a CI behaves like the 1D Kitaev p-wire and flips the fermion parity of a passing quantum vortex. We numerically verify this by putting the SC-CI model (4.11) on a 32× 32 periodic lattice. Similar to the previous case, we arrange a vortex anti-vortex pair and consider a circular vortex trajectory around the end of the SC trench where a MBS sits. The pairing phase ∆(r, t) = ∆x+ i∆y =|∆|eiϕ(r,t) is also given by (4.9). Figure4.5(b) shows
localized at the two vortex cores. There is a single level-crossing signifying the parity flip of the vortex as it travels across the SC trench. Figure 4.5(b) and (c) shows the localized wavefunctions of the zero energy MBS pair and the vortex state near the crossing respectively. The SC-CI setup is even more prefarable than the previous cases in demonstrating the vortex parity flip. Unlike the SC QSH-NI interface, the vortex excitation here stays localized at the vortex core throughout the evolution and never hybridizes with the bulk. This means that in the absence of accidental fermion poisoning, the vortex would carry a different electric charge after a complete cycle.
This non-interacting case which we could study explicitly shows us a way to understand the fermion parity flip AS. We can actually see signatures of the parity flip in terms of level crossings of the vortex states.